Design of Turbulent Flame Aerosol Reactors by Mixing-Limited Fluid

ARTICLE pubs.acs.org/IECR

Design of Turbulent Flame Aerosol Reactors by Mixing-Limited Fluid Dynamics Arto J. Gr€ohn,† Beat Buesser,† Jorma K. Jokiniemi,‡ and Sotiris E. Pratsinis*,† †

Particle Technology Laboratory, Institute of Process Engineering, Department of Mechanical and Process Engineering, Swiss Federal Institute of Technology (ETH) Zurich, CH-8092 Zurich, Switzerland ‡ VTT Technical Research Centre of Finland, Fine Particles, P.O. Box 1000, FI-02044 Espoo and Department of Environmental Science, Fine Particle and Aerosol Technology Laboratory, University of Eastern Finland, P.O. Box 1627, FI-70211 Kuopio, Finland

bS Supporting Information ABSTRACT: Nanoparticle synthesis in turbulent flame aerosol reactors is elucidated by computational fluid dynamics (CFD). Mixing-limited combustion is modeled, and total particle number, area, and volume concentration are described by transport equations including terms for particle dynamics. The spread of the particle size distribution at a given streamline is neglected as flame-made aerosols rapidly attain their self-preserving distribution. Results are in good agreement with primary particle data of turbulent diffusion flame synthesis of silica nanoparticles by oxidation of hexamethyldisiloxane vapor at different laboratories without adjustable parameters. Measured agglomerate mobility diameters best matched the predicted volume-equivalent softagglomerate diameters. The employed fractal-like dimensions (Df = 1.5-3) had no effect on the predicted primary particle and aggregate diameters and a rather small effect on volume-equivalent soft-agglomerate diameters.

’ INTRODUCTION Flame aerosol technology has become an increasingly attractive route for manufacturing sophisticated nanomaterials,1 well beyond today’s commodities2 of carbon black, pigmentary TiO2, and fumed SiO2. The advantages of this technology include no liquid byproduct requiring expensive cleaning, easy collection of particles from gases (rather than liquids), fewer process steps, high-purity products, unique filamentary morphology at high yields, which is attractive for nanocomposites, and the possibility to make metastable materials.3 Combined, these advantages make flame aerosol reactors suitable for large-scale production of nanomaterials.4 Understanding particle dynamics in flames allows control and, most importantly, facilitates design of new flame aerosol reactors.5 In these reactors certain particle characteristics (primary particle size, crystallinity, and extent of aggregation or hard-agglomeration) are fixed and cannot be changed by subsequent “finishing”, like surface functionalization, coating, or dispersion processes (e.g., pigmentary TiO2). As a result, understanding how to control the above particle characteristics is essential for the design and operation of aerosol reactors. By now, it is reasonably well understood that nanostructured particles are made in flames primarily by surface growth, coagulation, and sintering. Though surface growth seems to be a dominant route for soot or carbon black formation,6 coagulation determines flame-made ceramic particle dynamics, especially at high concentrations.7 To capture, however, the characteristic features of flame aerosol reactors for material synthesis, fluid and particle dynamics have to be followed in all spatial dimensions. Computational fluid dynamic (CFD) models of methane combustion were combined with Lagrangian descriptions of alumina8 r 2011 American Chemical Society

and titania9 monodisperse particle dynamics in diffusion flame reactors accounting for coagulation and sintering. Good agreement with measured particle sizes was obtained when an effective sintering rate was estimated by comparing simulated and measured particle surface areas.8,9 Titania nanoparticle synthesis in diffusion flame reactors was investigated also by Wang and Garrick.10 They simulated methane combustion and precursor oxidation with one-step Arrhenius rates in two-dimensional flames and particle size distribution dynamics by accounting for coagulation and surface growth. The mean particle diameter and geometric standard deviation increased with precursor concentration consistent with experimental data.11 Ji et al.12 simulated silica synthesis in hydrogen flame reactors by implementing moments of the size distribution into CFD without distinguishing between primary particles and agglomerates. Good agreement with measured particle diameters13 was found by adjusting particle formation and growth rates. Yu et al.14,15 investigated the effects of oxidant and precursor flow rates in diffusion flame reactors using the eddy-dissipation concept16 and accounting for nucleation, coagulation, sintering, and surface growth. Simulated flame temperatures were within 200 K of measurements, and good agreement with measured primary particle sizes was reported by adjusting sintering rate.17,18 Manenti and Masi19 designed novel tangential-based nozzles for flame reactors by CFD aerosol dynamics that reduce recirculation and broadening of the particle size distribution without distinguishing between primary and agglomerate particles. Received: August 25, 2010 Accepted: December 29, 2010 Revised: December 16, 2010 Published: February 01, 2011 3159

dx.doi.org/10.1021/ie1017817 | Ind. Eng. Chem. Res. 2011, 50, 3159–3168

Industrial & Engineering Chemistry Research

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Widiyastuti et al.20 included nucleation, coagulation, evaporation, and surface growth to study solid-fed flame synthesis of silica without defining separate primary particle and agglomerate diameter. Particle size was measured with field-emission scanning electron microscope and was in agreement with their simulations. Particle formation is often set identical to precursor oxidation rate defined by Arrhenius-type expressions,10,12,14,15,19 which are not usually available and may not be needed at the high temperatures of turbulent combustion when reactions are limited by mass transport. Furthermore, combustion is frequently simulated separately from particle formation that may be much slower than fuel oxidation. For turbulent nonpremixed combustion, a mixing-limited reaction model or mixture fraction theory21 can be used that allows solving chemical reactions and particle formation simultaneously, without using finite reaction rates. Using the mixture fraction theory, Sivathanu and Faeth22 showed that in hydrocarbon-air diffusion flames the local fuel equivalence ratio which is a function of mixture fraction can be used to predict species mass fractions and temperatures (within 200 K). Obieglo et al.23 found equally good agreement with measured species concentrations and temperatures for a hydrogen-air flame by assuming chemical equilibrium. Here such a model is interfaced with monodisperse aerosol dynamics elucidating formation of silica agglomerate and primary particles in turbulent diffusion flames for various silica sintering rates and fractal-like dimensions and compared to experimental data.24,25

’ THEORY Particle Formation by the Mixture Fraction Theory. In diffusion flames chemical reaction rates can be fast and limited by fuel/precursor (e.g., H2/SiCl4 or TiCl4) and oxidant (e.g., air or O2) mixing at the molecular level. Consequently, oxidation reactions and particle formation depend on reactant transport. When chemistry is fast, it is possible to obtain the spatial distribution of the precursor mass fraction by the mixture fraction theory (see Appendix). In this model the turbulent diffusivity is equal for all species even though they have different molecular diffusivities. This allows extracting the local particle formation rate, kf, by computing the mass flux imbalance in each grid cell

- 2NA X m j X H, j Vcell MH j ¼ 1 q

kf ¼

ð1Þ

where q is the number of cell faces, NA is Avogadro’s number, Vcell is the cell volume, MH is the precursor molecular weight, mj is the signed (positive for out- and negative for in-flow) mass flux through cell face j, and XH,j is the mass fraction of the precursor at cell face j. The -2 in eq 1 is needed as two SiO2 molecules are formed by oxidation of one hexamethyldisiloxane (HMDSO, C6H18OSi2) molecule here. Aerosol Dynamics. Precursor oxidation results in silica monomers (molecules) which can be regarded as particles.26 This approximation is reasonable here since the SiO2 vapor is highly supersaturated, and even if the critical particle size for nucleation consists of few molecules, coagulation quickly washes out initial differences27 for the employed SiO2 production rates (8-17 g/h). High particle concentrations and process temperatures lead typically to rapid consumption of a precursor (e.g., SiCl4, HMDSO) and attainment of the self-preserving aerosol

size distribution by coagulation.28 Then a single moment or monodisperse model29 is sufficient to predict reliably (within 20%)27 average aerosol properties (e.g., primary particle diameter). Recently, fumed SiO2 hard-agglomerates (aggregates) have been characterized in detail30 and found to consist of rather narrowly distributed primary particles as sintering narrows also their size distribution similar to condensation.31 This approach is used here so the evolution of total agglomerate number (N), area (A), and volume (V) concentration was modeled with CFD transport equations. implemented to ANSYS Fluent32 with user-defined functions as ∂ðui FNÞ ∂ μt ∂N 1 ð2Þ ¼ kf - βðFNÞ2 ∂xi ∂xi Sct ∂xi 2 ∂ðui FAÞ ∂ μt ∂A FðA - Nas Þ ð3Þ ¼ kf a0 ∂xi ∂xi Sct ∂xi τs ∂ðui FV Þ ∂ μt ∂V ð4Þ ¼ kf v0 ∂xi ∂xi Sct ∂xi The two left-hand side terms in eqs 2-4 describe particle convection and diffusion in turbulent flows. The μt/Sct is the diffusion coefficient (eq A2, Appendix), while v0 and a0 are the volume and surface area of a SiO2 monomer (molecule) with diameter 4.4 10-10 m, respectively. The coagulation rate, β, in the right-hand side of eq 2 was computed with the Fuchs interpolation kernel where the particle structure is taken into account by replacing the particle radius with the agglomerate radius of gyration.29 The fractal dimension, Df, used in the calculation of β was shown25 to depend mostly on O2 flow for different SiO2 production rates in the employed diffusion flame reactors.24 Accordingly, the Df for each O2 flow rate was obtained from experimental data25 (Df = 1.76-2.07) for all simulations. The variability of Df has been shown to affect the predicted agglomerate size but rather little the primary particle size.33 To study this sensitivity, simulations were conducted also with Df = 1.5 and 3. The total surface area (eq 3) is reduced by sintering of agglomerates at high temperatures. The characteristic sintering time, τs, is the time needed to reduce ∼63% of the excess surface area34 over that of a completely fused agglomerate, as. Here the performance of the following τs has been investigated35,36 4 8:310 ð5Þ τs ¼ 6:5 10 - 13 dp 3 e T

τs ¼ 6:5 10 - 13 dp 3 e

τs ¼

8:3104 T

3

-9

1 - 10dp

8 > > > > 0, >
> > > ¥, > :

d0 T < Tmelt 3 1 dp

ð6Þ

! !

ð7Þ

Equation 5 derived by Xiong et al.35 from the property data of Kingery et al.37 showed good agreement with the measurements of Seto et al.,38 who used it to predict sintering of SiO2 with an initial primary particle diameter, dp, of 10 nm in a hot-wall reactor. This expression was modified by Tsantilis et al.36 (eq 6) to include an additional term to account for the lower melting 3160

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temperature of the smallest nanoparticles (dp < 10 nm) than that of bulk silica.39 Equation 6 was used here unless otherwise indicated. An alternative approach has been presented recently by Heinson et al.,40 who divided the sintering process to separate full coalescence and fractal agglomeration regimes. By using such a two-step formulation, essential features of agglomeration could be captured without detailed knowledge of the sintering mechanics. This approach applies best to soft-agglomerates as it cannot account for sinter-necks of hard-agglomerates or aggregates. Accordingly, here an expression (eq 7) for sintering time is proposed where particles coalesce immediately above their melting temperature while when below no sintering or neck growth takes place. In eq 7 the bulk melting temperature Tmelt = 1986 K41 of SiO2 is multiplied with an additional term accounting for the lower melting temperature of the smallest primary particles.36 The minimum diameter used in the numerator of this term is defined as that of a monomer. Other expressions for τs have been presented offering good agreement with data by adjusting the pre-exponential factor and activation energy35 or the scalar values42 of eq 6 to account for the fluid mechanics in aerosol simulations. The primary particle diameter, dp, number of primary particles, np, and agglomerate collision diameter, dc, are29 6V A

ð8Þ

6V Nπdp3

ð9Þ

f dc ¼ dp n1=D p

ð10Þ

dp ¼ np ¼

Here the onset of soft agglomeration is defined as the point where dp has reached 95% of its final value. The mixing-cup averaged Sauter mean equivalent diameters as well as their standard deviation are obtained from the total fluxes of number (N), area (A), and volume (V) concentrations through radial slices along the reactor axis. Numerical Implementation. The reactor was modeled with a two-dimensional mesh using the axi-symmetric ANSYS Fluent32 solver with SIMPLEC pressure-velocity coupling.43 Figure 1 shows the geometry and structured grid, consisting of 58 927 cells. To model turbulent flow the realizable k-ε model44 (Appendix) was employed where pressure discretization was done with a second-order scheme.32 All field variables were solved with second-order upwind discretization.32 For the thermodynamic database, seven species were included (CH4, CO2, H2O, HMDSO, N2, O2, and SiO2). Intermediate species of methane combustion were excluded since no CO was observed in the Fourier transform infrared spectra (FTIR)45 of Mueller et al.24 The employed mesh was selected by simulations exploring optimal performance with respect to predicted temperature and precursor conversion.24 Particle formation is allowed only in cells where oxidation can take place, i.e., in cells where the equivalence ratio is above the lower flammability limit (0.4) for pure methane.46 The rich equivalence ratio flammability limit (1.6)46 was used to obtain the upper limit (0.6) for the mixture fraction. Gravity was included to account for buoyancy. Temperature was set to 300 K at all boundaries. The employed boundary conditions are presented in Table 1.

Figure 1. Axially symmetric simulation domain with boundary labels. The length of boundary 7 is 0.6 m, the radius is 0.07 m (boundary 8), and the burner length inside the reactor (boundary 6) is 0.03 m. The burner consists of three concentric tubes with a length of 0.13 m (boundary 4). The inner diameter of the center tube (boundary 3) is 4.8 mm, while the first and second annuli have inner-outer diameters of 5.6-6.4 (boundary 2) and 7.3-9 mm (boundary 1), respectively. Here the geometry is rotated 90 clockwise.

’ EXPERIMENTAL SECTION Mueller et al.24 used an enclosed methane/oxygen diffusion flame to produce silica nanoparticles for dental nanocomposites. Nanoparticles were formed by oxidation of HMDSO vapor fed with the CH4 stream at production rates of 9 and 17 g/h. The effect of the oxidant flow rate (1.3-24 L/min) on the produced particles was investigated focusing here on high O2 flow rates (>10 L/min) to be consistent with the employed combustion and fluid dynamics in fully turbulent flows. The flame temperatures and fuel (CH4 and HMDSO) conversion profiles were obtained by line-of-sight FTIR spectroscopy.45 The product primary particle diameter was measured by N2 adsorption and electron microscopy.24 In similar measurements, Scheckman et al.25 investigated, in addition to primary particle size, the structure and size of the product SiO2 agglomerates by combining a differential mobility analyzer (DMA) with an aerosol particle mass analyzer (APM) that led to direct measurement of the product silica fractal dimension. Both studies24,25 used the same radially symmetric burner consisting of three concentric tubes to supply the fuel N2/CH4/HMDSO, inert N2, and O2 streams in that order from the burner center to its outer annulus. ’ RESULTS AND DISCUSSION Comparison to Measured Temperature and Precursor Conversion. The model performance is evaluated first with

the FTIR45 temperature and precursor conversion measurements of Mueller et al.24 The comparison with temperature measurements is challenging since it is obtained by fitting a blackbody Planck function of a certain temperature with the normalized radiance spectra45 of hot CO2. Therefore, a CO2concentration weighted average line-of-sight temperature is computed from the simulations. Figure 2 shows the radial average (left) and CFD (right) axial evolution of (a) HMDSO and CH4 conversion and (b) temperature as measured (symbols)24 and simulated (lines) here at 22.7 (diamonds and solid line) and 13.3 L/min of O2 (squares and dashed line) at 17 g/h SiO2 production rate. Beyond 7 cm height above the burner (HAB) the predictions of temperature and fuel conversion rate are quite good for the more turbulent flame (22.7 L/min O2, Re = 4227). For the lower oxygen flow rate (13.3 L/min O2, Re = 2748), the prediction and measurements differ more, even though they follow consistent trends. The initial differences in measured and simulated temperature (HAB < 7 cm, 22.7 L/min O2, Figure 2b, left) may arise from the highly nonuniform radial temperature field which makes difficult the comparison of line-of-sight measurements with the CO2 3161



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Table 1. Boundary Conditions Used in the Simulationsa boundary 1. oxygen inlet

ref

composition vol. %

velocity inlet

8.73-18.38

O2: 100

Scheckman et al.25

velocity inlet

9.19-15.32

O2: 100

velocity inlet

1.11

N2: 100

velocity

(17 g/h) 3.96

(17 g/h) N2, 66.05; CH4, 32.60; HMDSO, 1.35

inlet

(9 g/h) 1.93

(9 g/h) N2, 65.19; CH4, 33.33; HMDSO, 1.48 (8.1 g/h) N2, 63.90; CH4, 34.90; HMDSO, 1.20

Mueller et al.24 Scheckman et al.25

a

velocity/pressure m 3 s-1/Pa

Mueller et al.24

2. nitrogen inlet 3. fuel inlet

type

velocity inlet

(8.1 g/h) 2.16

4. burner walls

wall no-slip

0

5. bottom wall 6. ambient inlet

wall no-slip pressure inlet

0 101 325

7. side wall

wall no-slip

0

8. outlet

pressure outlet

101 325

N2, 80; O2, 20

Conditions of Scheckman et al.25 and Mueller et al.24 for three SiO2 production rates are included. Temperature at all boundaries was 300 K.

Figure 2. Radially average measured24 and simulated (left): (a) fuel (CH4 þ HMDSO) conversion rates and (b) temperatures for oxygen flow rates 13.3 and 22.7 L/min at a SiO2 production rate of 17 g/h. On the right, detailed profiles of (a) HMDSO concentration and (b) temperature for a 22.7 L/min oxygen flow rate at a SiO2 production rate of 17 g/h.24 Note the difference in scale between these two profiles.

Figure 3. Production of 17 g/h of SiO2 by HMDSO oxidation with 22.7 L/min O2 in a diffusion flame aerosol reactor:24 (a) Particle formation rate in a magnified region above the burner with gas streamlines and (b) particle mass concentration per unit mass of gas.

averaged simulations. This is supported by Figure 2b (right), where as temperature becomes radially more uniform (HAB > 7 cm) better agreement with measurements is obtained. The present mixing-limited reaction model underestimates the fuel conversion rate when the flow becomes less turbulent as the HMDSO/CH4 conversion can become kinetically limited. Underestimation of the reaction rate leads also to underestimation of temperature (Figure 2b, left) due to the slower release of energy. On the basis of Figure 2, the oxygen flow rate of 13.3 L/ min can be considered as the lower limit where the model may still give realistic results. Particle Dynamics. Figure 3 shows the simulated (a) particle formation rate in a magnified (boxed in Figure 2b, right) region above the burner and (b) solid particle mass concentration profiles

per unit mass of gas in the reactor24 at 22.7 L/min O2 and SiO2 production rate of 17 g/h. Particle formation starts where oxygen and fuel streams mix (Figure 3a: HAB < 1 cm). High temperatures are observed in a relatively small reactor volume as has been seen also for alumina8 and titania9 production in such reactors without however employing Arrhenius reaction rates here. The more diffuse solid particle mass concentration field, which is almost equal to particle mass fraction, shows (Figure 3b) that particles traveling along their pathlines (Figure 3a) experience distinctly different temperatures and formation rates.8,9 This leads to a spatial variation in particle characteristics and possibly to a nonuniform product. Nevertheless, turbulence (Re = 4227) seems to facilitate the attainment of radially uniform particle mass concentration downstream (HAB = 35 cm) of the burner. 3162



Figure 4. Production of 17 g/h of SiO2 by HMDSO oxidation with 22.7 L/min O2 in a diffusion flame aerosol reactor:24 (a) primary particles per agglomerate, (b) primary particle diameter, and (c) agglomerate volume equivalent diameter.

Figure 4 shows the evolution of particle characteristics in the reactor at the conditions of Figure 3. Early in the process (close to the burner), the number of primary particles in an agglomerate, np (Figure 4a), depends strongly on temperature (Figure 2b). In the cooler edges of the flame, the sintering rate is lower, resulting in agglomerates that consist of many small primary particles.47 In contrast, at the center of the flame (HAB = 1-8 cm), where temperature quickly increases, particles fuse rapidly to spherical ones with a rather large diameter without distinction between primary and agglomerate particles.9 These particles do not become smaller further upstream as might be inferred erroneously from Figure 4b (e.g., the red becomes green or even blue from 13 cm onward). Higher up, the particles from all streamlines mix, causing the average primary particle diameter (Figure 4b) to decrease in the center (from red to green) and increase in the edges of the flame (from dark to light blue), respectively. Such large primary particles end up in the product powder collected on the filter as seen by microscopy, e.g., Scheckman et al.,25 Figure 4. In Figure 4c the volume-equivalent agglomerate diameter (dv) increases steadily with HAB by coagulation. The smallest dv can be observed early within the fringes of fuel/HMDSO and O2 jets (HAB < 4 cm) and at the edges of the flame where particle mass concentrations (Figure 3b) are lower than the center. Nevertheless, these radial differences in dv and other particle characteristics are smoothed by mixing downstream (HAB > 30 cm) and become constant across the reactor radius and eventually at its outlet as reported experimentally.47 Figure 5 summarizes the evolution of radially averaged Sauter mean primary particle (blue line) or agglomerate collision (red line) diameters along the burner axis for (a) 9 and (b) 17 g/h SiO2 production rates as well as their corresponding arithmetic standard deviation (shaded area). For both production rates, initial (HAB < 10 cm) radial differences in particle characteristics (Figure 4) result in large variability in both dc and dp. Further downstream

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Figure 5. Predicted evolution of radially averaged primary particle and agglomerate collision diameters with eq 6 for 22.7 L/min O2 at SiO2 production rates of (a) 9 and (b) 17 g/h.24 The vertical dotted lines indicate the onset of soft-agglomerate formation at the end of sintering, signified by the attainment of a practically constant primary particle diameter. The radial arithmetic standard deviation at each HAB is indicated with the shaded area.

(HAB > 10 cm), turbulent mixing starts to reduce the radial differences of dc and dp, reducing also their standard deviation. Initially agglomerate particles are formed containing many primary particles as can be expected from Figure 4a, where np increases immediately above the burner.8,9 Particles form agglomerates early on because their collision rate is much faster than their sintering rate while the flame temperature is still increasing. At increased HAB (1-4 cm), the average primary particle diameter increases while the agglomerate diameter decreases or remains rather constant. This is reflected by an overlap in the variability of dc and dp there. The radially averaged primary particle diameters increase continuously until sintering practically stops (dotted line at HAB approximately 4 or 7 cm for 9 or 17 g/h SiO2, respectively). There the largest hard-agglomerate particles are formed and from then on soft-agglomerates start to form.48 Here the numerical accuracy might cause fluctuations of ∼1% to the radially averaged values computed from the interpolated CFD fields. As a result, sintering was defined to end when dp reaches 95% of its final value instead of 99% used previously.48 Increasing particle production (and fuel flow rate)24 increased the high-temperature particle residence time and particle concentration, producing larger primary particles, consistent with experimental data.24 Nevertheless, the agglomerate size is smaller for the higher production rate until HAB = 15 cm because precursor oxidation ends higher up in the flame so aerosol concentrations are lower, resulting in slower agglomerate growth by coagulation at given HAB than at lower production rates. Comparison to Experimental Particle Size Data. Figure 6 compares the calculated (lines) and measured24 (symbols) 3163



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Figure 6. Measured24 and simulated primary particle diameters for two SiO2 production rates. Three sintering time expressions were used in the simulations for each data point.

radially averaged primary particle diameters, dp, for 9 (red squares) and 17 g/h (blue diamonds) SiO2 production rate as a function of oxygen flow rate for different sintering time expressions (eqs 5-7). Here increasing the oxygen flow rate and thus increasing aerosol dilution decreases flame temperature, particle concentration, and high-temperature particle residence time.24 As a result, agglomerates coagulate at a lower rate and sinter less so dp decreases when oxygen flow is increased. Reasonable agreement with data is obtained with the sintering time of Tsantilis et al.36 (eq 6), which reproduced the measured dp quite well at high O2 flow rates (>15 L/min). Reasonable agreement was found also with eq 5 for the 9 g/h SiO2. For the higher production rate, however, eq 5 predicts smaller dp in contrast to data. This is caused by precursor oxidation taking place higher up in the flame as formation of new monomers decreases the average dp. For example, at 22.7 L/min O2 flow SiO2 monomers are formed up to HAB = 8 cm for 17 g/h SiO2 production rate. At HAB = 8 cm the flame temperature is already decreasing (Figure 2b), which reduces the sintering time, thus hindering the coalescence of newly formed particles according to eq 5. Instead, the additional term in eq 6 accounting for the rapid coalescence of the smallest particles (dp < 10 nm) removes this sensitivity to the monomer formation. By assuming immediate coalescence above the particle melting temperature and no coalescence below according to eq 7, the dp could be predicted equally well as with eq 6 (Figure 6). Such simplification might be useful because only the melting temperature, which is a readily available material property, has to be known instead of the sintering kinetics for a given material. However, this method can be used only for predicting primary particle and soft-agglomerate size, while no information of hardagglomerate size or neck growth is obtained. The consistency of the present model is investigated further also with the measurements by Scheckman et al.25 at 8.1 g/h SiO2 production rate for Df = 1.5-3 in Figure 7a using eqs 6 and 7. Again, increasing O2 flow rate decreases primary particle diameters as seen in Figure 6. Also, the agreement of simulated (lines) and measured (circles) values is reasonably good without any adjustable parameters as above. The selected Df hardly makes a difference in model predictions,33 and eqs 6 and 7 are in reasonable agreement with the data as in Figure 6. For eq 5 the underprediction was more significant as in Figure 6 (not shown). Given the model assumption (monodisperse aerosol dynamics), excellent agreement between the present theory and data is obtained for high O2 flow rates where the fluid and combustion

Figure 7. Measured25 and simulated (a) primary particle and (b) agglomerate volume-equivalent diameters as a function of oxygen flow rate. The production rate of SiO2 was 8.1 g/h. Sintering time is from eqs 6 and 7, while Df is that measured25 at each O2 flow rate or Df = 1.5 or 3. The radial arithmetic standard deviations for hard- and soft-agglomerate diameters are approximately 30 and 70 nm, respectively. The shaded areas represent one standard deviation above and below the average: (a) primary particle diameters and (b) hard- (dark gray) and soft- (light gray) agglomerate diameters.

models for fully turbulent flows perform best. Furthermore, the geometric standard deviation measured by Scheckman et al.25 (Table 2) is 1.63-1.75 for Df = 1.76-2.4, which is consistent with the agglomerates’ theoretical49 self-preserving σg in the freemolecular regime of 1.52-1.6 for Df = 2-2.4. Figure 7b shows the predicted hard- and soft-agglomerate diameters by the model for Df = 1.5 and 3 and eq 6 as well as by the measured25 Df with eqs 6 and 7 along with the measured mobility diameter.25 Simulated hard- and soft-agglomerate sizes are obtained by dividing the total volume concentration flux with the total number concentration flux. The predicted SiO2 aggregate (or hard-agglomerate) diameter (red lines) corresponds to the agglomerate diameter when dp no longer changes by sintering (Figure 5). In Figure 7b the measured agglomerate mobility diameters were within the simulated hard- and soft-agglomerate diameters but closer to the soft one at high O2 flow rates where the model performs best. The predicted soft-agglomerate diameter (blue lines) in Figure 7b is extracted by doubling the sampling height25 of 11-22 cm to account for the length of the sampling funnel before the silica aerosol is diluted to enter the DMA.25 At such heights, radial variations in agglomerate diameter have not been removed by mixing, as shown in Figure 5, creating uncertainty which may partly explain the observed differences between measurements and the model. This may explain that even though agglomerates were seen by microscopy, very few of them were present in dental nanocomposites.24 Clearly, in these polymeric viscous suspensions, soft-agglomerates can be ruptured into their constituent primary and aggregate 3164


Industrial & Engineering Chemistry Research nanoparticles.50 Given that the aggregate size (red lines) is quite close to the primary particle diameter (circles, green lines) no large soft-agglomerates could be observed in polymers,50 though quite a few could be seen by microscopy when sampled above the flame or from the filter.25 In Figure 7 the predictions with eq 7 give similar results with eq 6 as shown previously in Figure 6, indicating that it is possible to predict primary particle and soft-agglomerate characteristics by using only the melting temperature. Also, the sensitivity of the results to variation in fractal dimension Df is shown. The lowest (Df = 1.5, dotted lines) and highest (Df = 3, solid lines) values were selected to cover the whole physically realistic range. These results show that the effect of Df on hard-agglomerate volumeequivalent and, especially, primary particle diameter is insignificant.33 Given that the standard deviation (e.g., Figure 5) for the soft-agglomerate diameter is (70 nm (light gray area in Figure 7b) while that of hard-agglomerate is (∼30 nm (dark gray area in Figure 7b), the measured agglomerate mobility diameters25 correspond better to soft-agglomerates, consistent with microscopy also.25

’ CONCLUSIONS The novelty of the presented method lies in its ability to predict particle formation rate, volume concentration of particles, and temperature in turbulent flame aerosol reactors for rapid gasto-particle conversion limited only by reactant mixing. Then the system defining the flame temperature and composition can be reduced to a single variable, the mixture fraction. Understanding this relation allows intuitive comprehension of the process; mixture fractions close to the stoichiometric value define where oxidation reactions and resulting energy release take place. Thus, the mixture fraction theory here was able to reasonably predict the measured radially averaged fuel conversion and temperature profiles without using kinetic reaction rates. The present CFD-based monodisperse aerosol model predicted well-measured SiO2 primary particle and agglomerate mobility diameters without any adjustable parameters (e.g., air entrainment dilution factor or effective sintering rate). It was shown also that primary particle and soft-agglomerate diameters can be predicted by using only the particle melting temperature by a new expression. The present model has the potential to facilitate reactor design for synthesis of particles with selected primary particle, agglomerate, and aggregate diameters. ’ APPENDIX Turbulent Combustion Model. The flow field was simulated by a realizable k-ε model44 which is based on the Reynoldsaveraged Navier-Stokes (RANS) approach. In RANS models the Navier-Stokes equations are decomposed into mean and fluctuating components. Time-averaged equations are derived by substituting the flow variables in the instantaneous continuity and moment equations with the decomposed terms. The turbulent kinetic energy and its dissipation rate are solved with model transport equations. The realizable k-ε model44 was selected based on performance evaluation of RANS models. In general, the standard51 and RNG52 k-ε models overestimated the reaction rates and underestimated the flame length possibly because of the round-jet anomaly.53 In contrast, the Reynolds stress model54 underestimated the reaction rate and flame temperature, while flame length was overestimated. All of the studied chemical reactions can be described by fast chemistry, so combustion is

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limited by the mixing of oxidizer and fuel streams (fuel here includes the nanoparticle precursor). A model for non-premixed turbulent combustion, implemented in a commercial CFD solver,32 was used here (Supplementary Information). It is based on mixture fraction theory22 where all species are assumed to be in chemical equilibrium. The mixture fraction f is defined as the mass fraction that originates from the fuel stream f ¼

Zh - Zh, ox Zh, f uel - Zh, ox

ðA1Þ

where Zh is the elemental mass fraction of species h, subscript “ox” denotes the oxidizer stream inlet (where f = 0), and subscript “fuel” denotes the fuel stream inlet (where f = 1). In the simulated diffusion flame, a nitrogen flow was introduced between oxygen and fuel streams as in experiments24,25 to prevent deposition of particles on the burner. This flow was included in the model as a secondary stream, modeled correspondingly with a secondary mixture fraction. The mixture fraction theory is based on the assumption that molecular diffusion is negligible compared to turbulent one, so diffusivities are equal for all species. Then reaction rate equations of individual species can be combined to a single transport equation since formation and decomposition source terms cancel out.21 Furthermore, eq A1 becomes identical to all species k, so the definition of f is unique. It follows that f is a conserved scalar, so its Favre-averaged transport equation can be written as32 ∂ μt ðFf Þ þ r 3 ð uBFf Þ ¼ r 3 rf ðA2Þ ∂t Sct where F is the gas density, u the gas velocity, and μt the turbulent eddy viscosity. In ANSYS Fluent a value of 0.85 is used for the turbulent Schmidt number Sct. Similar transport equations are solved for the mixture fraction variance as well as for the secondary mixture fraction and mixture fraction variance. The temperature, density, and species mass fractions can all be obtained as functions of f.21 To model the turbulence-chemistry interactions, probability density functions (PDF) are used to relate the instantaneous values of the fluctuating scalars to the averaged values. The reaction model is closed using an assumed shape PDF, here, namely, the double delta function, which is defined in terms of mixture fraction and its variance. A nonadiabatic extension of the reaction model was used since it is realistic to assume that the inlet gas temperature increases by the heating of the burner. In this extension the influence of enthalpy to the thermochemical state is added to the equations defining temperature and species mass fractions. The energy equation was solved to account for heat transport inside and through the domain boundaries. Thermodynamic Database. To compute the species mass fractions and other scalar values in the chemical equilibrium model, a thermodynamic database is used which contains temperature (T) dependent values of heat capacity (Cp), entropy (S), and enthalpy (H) for each species (k). The database used by ANSYS Fluent is in the Chemkin format55, where variables are stored using polynomial coefficients (γi) given as (Table A1) Cop, k R 3165

¼ γ1 þ γ2 T þ γ3 T 2 þ γ4 T 3 þ γ5 T 4

ðA3Þ



ARTICLE

Table A1. Polynomial Coefficients (γi) Used in Equations A3-A5a coefficient

SiO2

γ1

a

HMDSO 1.16 101

3.45

γ2 γ3

-3

7.88 10 -6.17 10-6

6.39 10-2 -1.32 10-5

γ4

2.16 10-9

-1.20 10-8

-13

γ5

-2.80 10

5.22 10-12

γ6

-3.71 10

-9.71 104

γ7

1.07 10

-9.51

4

1

Same values are used for the high- and low-temperature regimes of the Chemkin database to ensure the continuity of the functions.

Sok γ γ γ ¼ γ1 ln T þ γ2 T þ 3 T 2 þ 4 T 3 þ 5 T 4 þ γ7 R 2 3 4

ðA4Þ

Hko γ γ γ γ γ ¼ γ1 þ 2 T þ 3 T 2 þ 4 T 3 þ 5 T 4 þ 6 RT 2 3 4 5 T

ðA5Þ

Here, R is the gas constant. The precursor (HMDSO) and product (SiO2) were added to the database using table values of ), and entropy heat capacity, heat of formation at 298 K (ΔfHk,298 ).56-58 The polynomial coefficients, of gas at 298 K (Sk,298 presented in Table A1, were used in eqs A3-A5, while entropy and enthalpy were approximated with55 Z Tk o Cp, k ∂T ðA6Þ Sok ¼ Sok, 298 þ 298 T and Hko

Z ¼ 0

Tk

Cop, k ∂T

o Δf H298

Z þ

Tk 298

Cop, k ∂T

ðA7Þ

It is noteworthy that values of only three common scalars (C, p S, and H) are needed to include a component into the combustion model.

’ ASSOCIATED CONTENT

bS

Supporting Information. Detailed listing of the employed computer codes. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*Tel.: þ41-44-632-3180. Fax: þ41-44-632–1595. E-mail: pratsinis@ ptl.mavt.ethz.ch.

’ ACKNOWLEDGMENT The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/ 2007-2013) under grant agreement n 228885. Financial support from a scholarship of the University of Kuopio, Fortum Foundation # 07-057 and Swiss National Science Foundation (SNF) grant # 200021-119946/1 are gratefully acknowledged. ’ NOMENCLATURE as = area of a completely fused aggregate (m2) a0 = surface area of SiO2 monomer (m2)

A = total agglomerate area concentration (m2/kggas) β = Fuchs coagulation kernel (m3/s) Cp = heat capacity (erg 3 mol-1 3 K-1) ΔfHk,298 = heat of formation at 298 K (erg/mol) dc = agglomerate collision diameter (m) dp = primary particle diameter (m) dv = agglomerate volume equivalent diameter (m) do = diameter of SiO2 monomer (m) Df = fractal dimension f = mixture fraction γi = Chemkin polynomial coefficients h = species index H = enthalpy (erg/mol) kf = particle formation rate (s-1 3 m-3) μt = turbulent eddy viscosity (kg 3 m-1 3 s-1) m = mass flux (g/s) MH = molecular weight of the HMDSO (g/mol) np = number of primary particles N = total agglomerate number concentration (1/kggas) NA = Avogadro’s number (mol-1) q = number of cell faces F = gas density (kg/m3) R = gas constant (erg 3 mol-1 3 K-1) Re = Reynolds number σg = geometric standard deviation Sct = turbulent Schmidt number S = entropy (erg 3 mol-1 3 K-1) Sk,298 = entropy of gas at 298 K (erg 3 mol-1 3 K-1) τs = characteristic sintering time (s) tres = residence time of particles in reactor (s) t = time (s) T = temperature (K) Tmelt = bulk melting temperature of SiO2 (K) u = gas velocity (m/s) v0 = volume of SiO2 monomer (m3) V = total agglomerate volume concentration (m3/kggas) Vcell = volume of computational cell (m3) xi = spatial position vector (m) XH,j = mass fraction of HMDSO at cell j Z = elemental mass fraction

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Design of Turbulent Flame Aerosol Reactors by Mixing-Limited Fluid

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